Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media

Yves Capdeboscq; Shaun Chen Yang Ong

doi:10.3934/dcdsb.2020228

Article Contents

2020, Volume 25, Issue 10: 3857-3887. Doi: 10.3934/dcdsb.2020228

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Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media

Yves Capdeboscq^1, and
Shaun Chen Yang Ong^2,

1.
Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France
2.
Mathematical Institute, University of Oxford, OX2 6GG, UK

Received: May 31, 2019

Revised: March 31, 2020

Early access: July 2020

Published: October 2020

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Abstract

We consider the conductivity equation in a bounded domain in $ \mathbb{R}^{d} $ with $ d\geq3 $. In this study, the medium corresponds to a very contrasted two phase homogeneous and isotropic material, consisting of a unit matrix phase, and an inclusion with high conductivity. The geometry of the inclusion phase is so that the resulting Jacobian determinant of the gradients of solutions $ DU $ takes both positive and negatives values. In this work, we construct a class of inclusions $ Q $ and boundary conditions $ \phi $ such that the determinant of the solution of the boundary value problem satisfies this sign-changing constraint. We provide lower bounds for the measure of the sets where the Jacobian determinant is greater than a positive constant (or lower than a negative constant). Different sign changing structures where introduced in [9], where the existence of such media was first established. The quantitative estimates provided here are new.

Erratum: The name of the second author has been corrected from Haun Chen Yang Ong to Shaun Chen Yang Ong. We apologize for any inconvenience this may cause.

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Mathematics Subject Classification: Primary: 35B30, 35B05; Secondary: 35B27, 35R30.

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References

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